Generalized Order Statistics

[SumitG]

Hi! Does anybody know about generalized order statistics? Apparently I'm supposed to have extensive knowledge on them for research this summer...but I'm having a very difficult time understanding it. I'm trying to gain some understanding from an article about it...but am having little luck. A quote about the concept from this article (titled Numerical and Graphical Data Summary Using 0-Statistics and by Kaigh and Driscoll if you're curious):

Let [unparseable or potentially dangerous latex formula] denote the usual order statistics from a sample [unparseable or potentially dangerous latex formula]. Then the O-statistics of order [unparseable or potentially dangerous latex formula] are defined by the equation

[unparseable or potentially dangerous latex formula]

The article states that this formula for [unparseable or potentially dangerous latex formula] is simply the expected value of the probability distribution for the probability [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] is the normal order [unparseable or potentially dangerous latex formula] statistic from a subsample of size [unparseable or potentially dangerous latex formula] from a sample of size [unparseable or potentially dangerous latex formula].

Besides this, the article does not mention how generalized order statistics are obtained at all, or why they are obtained as such. So, I'm having problems with these ideas, as well as the elementary combinatorial argument that is needed to obtain that formula. Any aid in understanding of this matter would be greatly appreciated. Thanks in advance.

[AuSmith]

Well, for the "elementary combinatorial argument," I don't think the x's are very variable anymore, so I will go with [unparseable or potentially dangerous latex formula] for [unparseable or potentially dangerous latex formula].

The problem reduces to showing if you pick [unparseable or potentially dangerous latex formula] numbers out of [unparseable or potentially dangerous latex formula], the probability that the [unparseable or potentially dangerous latex formula]th smallest number will be [unparseable or potentially dangerous latex formula] is the probability of three events all occuring:

Of our [unparseable or potentially dangerous latex formula] numbers,

1) exactly [unparseable or potentially dangerous latex formula] of them must land in [unparseable or potentially dangerous latex formula],

2) the [unparseable or potentially dangerous latex formula]th one must be [unparseable or potentially dangerous latex formula], and

3) exactly [unparseable or potentially dangerous latex formula] of them must land in [unparseable or potentially dangerous latex formula]

So, since each of the [unparseable or potentially dangerous latex formula] combinations is equally likely, we only need to count how many of them satisfy each of the above three events. But, any two of these events implies the third, and any two of them are independent (check). The next question would then be, "how many ways?"

[unparseable or potentially dangerous latex formula]

There you go?

[SumitG]

Thanks a lot! It's a great deal of help. I can't believe I couldn't get that though...just shows how much work I need to do on counting...

[SumitG]

Hooray! After your post, and reading the paper over about 40 times, I think I've finally got the ideas behind gOS. Thanks again.